nLab
cotangent bundle

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

Given a differentiable manifold XX, the cotangent bundle T *(X)T^*(X) of XX is the dual vector bundle over XX dual to the tangent bundle TxT x of XX.

A cotangent vector or covector on XX is an element of T *(X)T^*(X). The cotangent space of XX at a point aa is the fiber T a *(X)T^*_a(X) of T *(X)T^*(X) over aa; it is a vector space. A covector field on XX is a section of T *(X)T^*(X). (More generally, a differential form on XX is a section of the exterior algebra of T *(X)T^*(X); a covector field is a differential 1-form.)

Given a covector ω\omega at aa and a tangent vector vv at aa, the pairing ω,v\langle{\omega,v}\rangle is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for T *(X)T^*(X) to be the dual vector bundle to T *(X)T_*(X). More globally, given a covector field ω\omega and a tangent vector field vv, the paring ω,v\langle{\omega,v}\rangle is a scalar function on XX.

Given a point aa in XX and a differentiable (real-valued) partial function ff defined near aa, the differential d af\mathrm{d}_a f of ff at aa is a covector on XX at aa; given a tangent vector vv at aa, the pairing is given by

d af,v=v[f], \langle{\mathrm{d}_a f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions defined near aa. (It is really the germ at aa of ff that matters here.) More globally, given a differentiable function ff, the de Rham differential df\mathrm{d}f of ff is a covector field on XX; given a vector field vv, the pairing is given by

df,v=v[f], \langle{\mathrm{d}f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions.

One can also define covectors at aa to be germs of differentiable functions at aa, modulo the equivalence relation that d af=d ag\mathrm{d}_a f = \mathrm{d}_a g if fgf - g is constant on some neighbourhood of aa. In general, a covector field won't be of the form df\mathrm{d}f, but it will be a sum of terms of the form hdfh \mathrm{d}f. More specifically, a covector field ω\omega on a coordinate patch can be written

ω= iω idx i \omega = \sum_i \omega_i\, \mathrm{d}x^i

in local coordinates (x 1,,x n)(x^1,\ldots,x^n). This fact can also be used as the basis of a definition of the cotangent bundle.

Properties

Symplectic structure

Every cotangent bundle T *XT^\ast X carries itself a canonical differential 1-form

θΩ 1(T *X) \theta \in \Omega^1(T^* X)

with the property that under the isomorphism

j:Γ(T *X)Ω 1(X) j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)

between differential 1-forms and smooth sections of the cotangent bundle we have for every smooth section σΓ(T *X)\sigma \in \Gamma(T^* X) the identification

σ *θ=j(σ) \sigma^* \theta = j(\sigma)

between the pullback of θ\theta along σ\sigma and the 1-form corresponding to σ\sigma under jj.

This unique differential 1-form θΩ 1(T *X)\theta \in \Omega^1(T^* X) is called the Liouville-Poincaré 1-form or canonical form or tautological form on the cotangent bundle.

The de Rham differential ωdθ\omega \coloneqq d \theta is a symplectic form. Hence every cotangent bundle is canonically a symplectic manifold.

On a coordinate chart n\mathbb{R}^n of XX with canonical coordinate functions denoted (x i)(x^i), the cotangent bundle over the chart is T * n n× nT^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n with canonical coordinates ((x i),(p j))((x^i), (p_j)). In these coordinates the canonical 1-form is (using Einstein summation convention)

θ=p idx i \theta = p_i d x^i

and hence the symplectic form is

ω=dp idq i. \omega = d p_i \wedge d q^i \,.

Revised on November 23, 2017 07:48:04 by Urs Schreiber (94.220.49.59)